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Name ID MATH 251 Sections 506 Exam 1 Spring 2006 P. Yasskin Multiple Choice: (5 points each. No part credit.) 1-9 /45 10 /20 11 /15 12 /10 13 /15 Total /105 1. A fly travels along the path ⃗ rt = 3 sin t, −4 sin t, 5 cos t. Find the arc length traveled by the fly between 3, −4, 0 and 0, 0, −5. a. π b. c. d. e. 2 π 3π 2 2π 5π 2 ̂ to the curve ⃗rt = 3 sin t, −4 sin t, 5 cos t. 2. Find the unit binormal B a. b. c. d. e. 3 , −4 , 0 5 5 3 , 4 ,0 5 5 4 , −3 , 0 5 5 4 , 3 ,0 5 5 −3 , 4 , 0 5 5 1 3. Find the vector projection of the vector ⃗ u = ⟨1, −2, 2⟩ onto the vector ⃗v = ⟨−1, 1, 1⟩. a. b. c. d. e. −1 , 1 , 3 3 1 , −1 , 3 3 1 , −1 , 9 9 −1 , 2 , 9 9 1 , −2 , 9 9 1 3 −1 3 −1 9 −2 9 2 9 4. Find the plane parallel to 4x − 3y + 2z = 5 which passes through the point P = 1, 2, 3. What is the z-intercept of this plane? a. 2 b. 4 c. 6 d. 8 e. 10 5. y The plot at the right is the contour plot of which function? a. The hyperbolic paraboloid z = x 2 − y 2 4 2 b. The hyperbolic paraboloid z = xy c. The hyperboloid z= 1 + x2 + y2 d. The cone z= x − 1 2 + y 2 e. The elliptic paraboloid z = x 2 + y − 1 2 -4 -2 2 4 x -2 -4 2 6. If r = x 2 + y 2 , find ∂ 2 r 3, 4. ∂x∂y a. 12 b. c. d. e. 5 12 25 12 125 −12 125 −12 5 7. The graph of a function z = fx, y passes through the point 2, 4, 8 and has f x 2, 4 = −3 and f y 2, 4 = 5. Use the linear approximation to approximate f2. 2, 3. 9 a. −1. 1 b. 1. 1 c. 6. 9 d. 7. 9 e. 9. 1 8. Duke Skywater is travelling through the galaxy. At the present time he is at the point with galactic coordinates P = 40, 25, 53 (in lightyears), and his velocity is ⃗v = . 1, −. 2, . 3 (in lightyears /year). He measures the polaron density to be U = 4300 polarons/cm 3 and its gradient to be ⃗ ∇U = 3, 2, 1 dU 3 polarons/cm /lightyear. Find the rate at which Duke sees the polaron density changing (in dt 3 polarons/cm /year). a. 0. 2 b. 1. 0 c. 223 d. 223. 2 e. 224 3 9. The density of carbon monoxide in a room is given by δ = z ln9 + x 2 − y 2 . If you start at the point 3, 4, 2, in what direction should you move to decrease the carbon monoxide density as fast as possible? a. −6, −8, − ln 2 b. −6, 8, − ln 2 c. 6, −8, − ln 2 d. −6, 8, ln 2 e. 6, −8, ln 2 Work Out: (Points indicated. Part credit possible. Show all work.) 10. Find the equation of the plane tangent to each of the following surfaces: a. 10 points z = fx, y = xe xy b. 10 points xe yz + ye xz = 5 at at x, y = 2, 0 x, y, z = 2, 3, 0 4 11. 12. The radius of a cylinder is currently r = 5 cm and its height is h = 10 cm. cm cm Its radius is increasing at dr = 0. 3 and its height is decreasing at dh = −0. 2 . dt dt sec sec Is its volume increasing or decreasing and at what rate? 15 points 10 points Find ∂z ∂v If z = x 2 y 3 where x = xu, v and y = yu, v satisfy ∂x 3, 4 = 5 ∂x 3, 4 = 6 x3, 4 = 2 ∂u ∂v ∂y ∂y y3, 4 = 1 3, 4 = 7 3, 4 = 8 ∂u ∂v u,v=3,4 . 5 13. 15 points Find the point on the ellipsoid f = −x − 2y + 4z is a minimum. 2 x 2 + y + z 2 = 3 at which the function 16 4 You may use any method but Lagrange multipliers is easier than eliminating a variable. 6